 Register   ## BAR AND PLATE STABILITY PROBLEMS 4/24/2015

Algorithms and methods of SP LIRA 10.4 computational processor are presented in the article. Displacements expressions across bar cross section and plate thickness for geometrically nonlinear problems are used.

BAR AND PLATE STABILITY PROBLEMS

Evzerov I.D., Doctor of Technical Sciences

Annotation
Bars and plates stability problems are considered. Variational formulations of stability problem are used. Positive definiteness of functional of potential energy is analyzed. Transition from three-dimensional stability problem to corresponding bar and plate problems is completed. Displacements expressions across bar cross section and plate thickness for geometrically nonlinear problems are used. These expressions derived from assumption, that bar in-plane deformations or deformations across plate thickness are equal to zero. The second variations of nonlinear deformations have been calculated. Integration over bar cross section and plate thickness has been done. Well known formulas for internal forces and equilibrium equation have been implemented. Stability functionals has been obtained for bars and plates. Comparison with previuosly known results has been made. Solution of test problem of axially loaded cantilever with cross section of Pi, which has been modeled using plates, is provided.

Author's notes
In the work presented here contains mathematical justification of solution methods of geometrically nonlinear problems, which are applied in SP LIRA 10. Bar's stability theory has been discussed with professor Slivker V.I., Doctor of Technical Sciences, and matches results obtained via other methods in monograph Perelmuter A.V., Slivker V.I. Structural equilibrium stability and related problems. - M.: SCAD SOFT, 2009.

Annotation
Bars and plates stability problems are considered. Variational formulations of stability problem are used. Positive definiteness of functional of potential energy is analyzed. Transition from three-dimensional stability problem to corresponding bar and plate problems is completed. Displacements expressions across bar cross section and plate thickness for geometrically nonlinear problems are used. These expressions derived from assumption, that bar in-plane deformations or deformations across plate thickness are equal to zero. The second variations of nonlinear deformations have been calculated. Integration over bar cross section and plate thickness has been done. Well known formulas for internal forces and equilibrium equation have been implemented. Stability functionals has been obtained for bars and plates. Comparison with previuosly known results has been made. Solution of test problem of axially loaded cantilever with cross section of Pi, which has been modeled using plates, is provided.

Key words: stability problems, bars and plates, variational formulations.
Key words: stability problems, bars and plates, variational formulations.

Introduction
Structural stability analysis os one of the major stages of the calculation. Bar and plates elements require particular attention. Equations for bars and plates are obtained from three-dimensional problem, using displacements expressions across bar cross section and plate thickness. The Euler–Bernoulli hypotheses are applied for bars [1,2], straight lines normal hypotheses for plates and shells [3,4], small parameter expansion methods [5,6,7,8] and other asymptotic methods [9,10,11]. Stability of bar of variable section is analyzed in [12,13,14,15]. Various examples of errors, that occur during stability analysis, are presented in [16,17].